|
%I #19 Sep 05 2021 22:07:09
%S 6,30,60,210,210,180,630,330,1320,1560,2340,990,2730,840,4620,3570,
%T 5610,4290,1710,7980,2730,6630,10920,12540,4080,8970,14490,18480,9690,
%U 3900,11550,25200,26910,30600,34650,32130,37050,7980,23460,6090,29580,49140,35700
%N a(n) = area of Pythagorean triangle with hypotenuse p, where p = A002144(n) = n-th prime == 1 (mod 4).
%C Pythagorean primes, i.e., primes of the form p = 4k+1 = A002144(n), have exactly one representation as sum of two squares: A002144(n) = x^2+y^2 = A002330(n+1)^2+A002331(n+1)^2. The corresponding (primitive) integer-sided right triangle with sides { 2xy, |x^2-y^2| } = { A002365(n), A002366(n) } has area xy|x^2-y^2| = a(n). For n>1 this is a(n) = 30*A068386(n).
%H T. D. Noe, <a href="/A145010/b145010.txt">Table of n, a(n) for n = 1..1000</a>
%F a(n) = A002365(n)*A002366(n)/2.
%F a(n) = x*y*(x^2-y^2), where x = A002330(n+1), y = A002331(n+1).
%e The following table shows the relationship between several closely related sequences:
%e Here p = A002144 = primes == 1 (mod 4), p = a^2+b^2 with a < b;
%e a = A002331, b = A002330, t_1 = ab/2 = A070151;
%e p^2 = c^2+d^2 with c < d; c = A002366, d = A002365,
%e t_2 = 2ab = A145046, t_3 = b^2-a^2 = A070079,
%e with {c,d} = {t_2, t_3}, t_4 = cd/2 = ab(b^2-a^2).
%e ---------------------------------
%e p a b t_1 c d t_2 t_3 t_4
%e ---------------------------------
%e 5 1 2 1 3 4 4 3 6
%e 13 2 3 3 5 12 12 5 30
%e 17 1 4 2 8 15 8 15 60
%e 29 2 5 5 20 21 20 21 210
%e 37 1 6 3 12 35 12 35 210
%e 41 4 5 10 9 40 40 9 180
%e 53 2 7 7 28 45 28 45 630
%t Reap[For[p = 2, p < 500, p = NextPrime[p], If[Mod[p, 4] == 1, area = x*y/2 /. ToRules[Reduce[0 < x <= y && p^2 == x^2 + y^2, {x, y}, Integers]]; Sow[area]]]][[2, 1]] (* _Jean-François Alcover_, Feb 04 2015 *)
%o (PARI) forprime(p=1,499, p%4==1 | next; t=[p,lift(-sqrt(Mod(-1,p)))]; while(t[1]^2>p,t=[t[2],t[1]%t[2]]); print1(t[1]*t[2]*(t[1]^2-t[2]^2)","))
%o (PARI) {Q=Qfb(1,0,1);forprime(p=1,499,p%4==1|next;t=qfbsolve(Q,p); print1(t[1]*t[2]*(t[1]^2-t[2]^2)","))} \\ _David Broadhurst_
%Y Cf. A002144, A002365, A002366, A144954.
%K nonn
%O 1,1
%A _M. F. Hasler_, Feb 24 2009
|
